Page 115 - 48Fundamentals of Compressible Fluid Mechanics

5.1. SOLUTION OF THE GOVERNING EQUATIONS 77

Rearranging equation (5.11) reads

(5.12)

Energy equation (5.3) converted to a dimensionless form as

(5.13)

It can be also noticed that equation (5.13) means that the stagnation temperature

because
is identical to

is the same,

. Under perfect gas model

(5.14)

Using the identity (5.14) transforms the momentum equation (5.2) into
(5.15)

Rearranging the equation (5.15) yields

(5.16)

The pressure ratio in equation (5.16) can be interpreted as the loss of the static

pressure. The loss of the total pressure ratio can be expressed by utilizing the

relationship between the pressure and total pressure (see equation (4.11)) as

(5.17)

%

is needed to be solved from the above

%

equations set. This relationship can be obtained from the combination of mass,

and
The relationship between the
momentum, and energy equations. From the equations (5.13) (energy) and equa-

tion (5.12) (mass) the temperature ratio can be eliminated.

(5.18)

%

Combining the results (5.18) with equation (5.16) results

%

(5.19)

%

%